The AdS/CFT correspondence [1]}
has been intensively investigated in the large \(N\)  limit,
and agreement of many quantities calculated on the both side of the duality has been confirmed.
Although the quantum gravitational effect becomes important for finite \(N\) ,
it may be possible to calculate some quantities associated with topology and supersymmetry
on the AdS side.
Let us consider the duality between 4d \({\cal N}=4\)  SYM and the string theory in \(AdS_5\times S^5\) .
The relation \(L^4/l_p^4\sim T_{\rm D3}L^4\sim N\)  among
the AdS radius \(L\) , the ten-dimensional Planck length \(l_p\) ,
and the D3-brane tension \(T_{\rm D3}\) 
suggests that when \(N\)  is finite not only the quantum gravitational effect
but also the contribution of D3-branes extended in \(AdS_5\times S^5\)  becomes important,
and the finite \(N\)  corrections for some quantities may be reproduced as the
contribution of D3-branes.
One such example is the BPS partition function of \({\cal N}=4\)  SYM.
It was calculated by the geometric quantization of BPS D3-brane configurations (sphere giants
[2]}, [3]})
in \(S^5\)  [4]}.
(See also [5]} for another derivation using AdS giants
[6]}, [7]}.)
In this paper we discuss a similar calculation of the superconformal index [8]}
for finite \(N\)  on the AdS side.
